How can I subdivide an edge at a specific point on an axis?
You can edge slide a single vertex.  Select it and type GG and then move it:

But if the sloped line has endpoints that have the same x coordinates as the bottom line, there's another way:
Subdivide the top line into the same number of segments as the bottom line:

Each new vertex in the sloped line has the same X coordinate as the corresponding vertex of the flat line.
There are various ways to select the vertices you want to remove.  One that works is to select the vertex you want to keep and then use CTRL+I to reverse the selection:

next shift-left-click on each of the end vertices to deselect them:

Then type x and select 'dissolve vertices' to get rid of the excess:

and you're left with what you want:

Notice that I had both lines share an endpoint.  This isn't necessary, and what I suggest would work in your example, so long as the two endpoints have the same X value